| | 156 | === Normalization === |
| | 157 | |
| | 158 | Now the final spectra [[latex($P_k$)]] has the property that [[latex($\sum(P_k) = <|f|^2>$)]] where k is normalized to [[latex($\frac{2\pi}{\max(L)}$)]] |
| | 159 | |
| | 160 | Now normally, [[latex($F(\mathbf{k}')$)]] would have units of [[latex($f(\mathbf{x}')V$)]] and [[latex($P_{3D}(\mathbf{k}')$)]] would have units of [[latex($f(x')^2 V^2$)]] |
| | 161 | |
| | 162 | Then to get the total power, you would integrate [[latex($E=\int P_{3D}(\mathbf{k}) \mathbf{dk}$)]] which would have units of [[latex($f(x')^2 V$)]] |
| | 163 | |
| | 164 | So if [[latex($f(x')=\sqrt{\rho v}$)]] then [[latex($E$)]] would have units of Kinetic Energy. |
| | 165 | |
| | 166 | Also, [[latex($P_{1D}(k)$)]] will have units of [[latex($f(x')^2 V L$)]] |
| | 167 | |
| | 168 | Because of our normalization of the FFT, [[latex($P_{3D}(k)$)]] has units of [[latex($f(x')^2$)]] and because of the way we bin the data, [[latex($P_{1D}(k)$)]] also has units of [[latex($f(x')^2$)]]. |
